The microspheres are analysed with a 100 keV design energy interferometer in order to understand the contributions to the scattering signal given by fine structures.
This work is an attempt to reproduce and extend the experiments done by Lynch et al. (2011) on the interpretation of dark-field contrast with respect to particle size.
We start from formula (68) of that paper for a monochromatic beam: \[ \mu_d = \frac{3\pi}{\lambda^2}f |\Delta\chi|^2 d \begin{cases} D' & \text{if } D' \leq 1\\ D' - \sqrt{D'^2 - 1}(1 + D'^{-2}/2) + (D'^{-1} + D'^{-3} / 4)\log\left(\frac{D' + \sqrt{D'^2 - 1}}{D' - \sqrt{D'^2 - 1}}\right) & \text{otherwise} \end{cases} \] Where \( \mu_d = -\log B/t \) with \(B\) as the dark-field signal and \(t\) the sample thickness. \(D'\) is a normalized particle size equal to \(Dp/\lambda L\) with \(L\) the sample to detector distance, \(p\) the period of \(G_2\) and \(D\) the particle size. \(\chi\) is the complex refractive index.
Instead of \(\mu_d\) we will use a quantity that is proportional to that through the absorption coefficient \(\mu = 2k\beta\), but is more readily accessible from the experimental data, that is the ratio of the logarithms of the dark field and the transmission. \[ R = \frac{\log B}{\log A} \]
In the case of a polychromatic beam, this formula needs to be computed for each energy bin \(\mathcal{E}\) of the spectrum and then summed over the spectrum: \[ R = \frac{\sum_\mathcal{E} w(\mathcal{E}) \log B(\mathcal{E})}{\sum_\mathcal{E} w(\mathcal{E}) \log A(\mathcal{E})} \]
The spectral weights \(w(\mathcal{E})\) are computed starting from a SpekCalc simulated spectrum. The result of the simulation can be downloaded as a csv file. The spectrum is then attenuated according to the Beer-Lambert law as it goes through the sample and interacts with the detector. This reweighting is performed by a python script. The refractive index \(n\) needed for this reweighting and for the function above is calculated through this library.
The visibility for each energy bin is calculated according to Thüring and Stampanoni (2014).
The formula for the visibility at an energy \( \mathcal{E} \) with a \(\pi\) shift in the phase grating, a design energy \(\mathcal{E_0}\) and a Talbot order \(m\) is \[ v(\mathcal{E}) = \frac{2}{\pi} \left\lvert \sin^2 \Big(\frac{\pi}{2}\frac{\mathcal{E_0}}{\mathcal{E}}\Big) \sin \Big(\frac{m\pi}{2}\frac{\mathcal{E_0}}{\mathcal{E}}\Big) \right\rvert . \]
This visibility as a function of energy is shown in the theoretical visibility plot below.
After the spectral weights are computed, and they are obviously different for different thicknesses, the function \(R\) can be fitted to the data with two free parameters \(C\) and \(R_0\) such that: \[ log B(\mathcal{E}) / t = C |\Delta n|^2 \mathcal{E} u(\mathcal{E}) \] \[ R = R_0 + C \frac{\sum |\Delta n|^2 \mathcal{E} u(\mathcal{E})}{\sum \mathcal{E}\beta u(\mathcal{E})} \] Where \(u(\mathcal{E})\) is the conditional statement of (68). \(C\) is a normalization parameter and \(R_0\) can be interpreted as the intensity of the dark-field signal without any substructure in the sample.
A nonlinear fit function in R is used. The results of the fit are shown in the plot on top and in the following table:
| sample | \(R_0\) | \(C (\times 10^{3})\) |
|---|---|---|
| sample thickness 12 mm | 0.746 ± 0.12 | 8.7e-09 ± 1.7e-09 |
| sample thickness 45 mm | 0.256 ± 0.11 | 2.6e-08 ± 2.9e-09 |
| sample thickness 12 mm source filter 1 mm Cu | 0.242 ± 0.13 | 2.6e-08 ± 3.7e-09 |
A previous experiment performed with samples without features on the micrometer scale, and a very similar setup with 120 keV design energy, showed that the ratio of the logarithms is a nonzero value and it is almost completely independent of the material. The values between 2 and 3 obtained with the fit on the microspheres are consistent with these earlier values and can provide a justification for merging these two regimes into one understanding of the dark-field signal.
With this experiment we show that we can understand the dark-field signal in the context of high-energy grating interferometry as given by two contributions:
The plots below show all the steps that lead to the final weighting of the spectrum. The final weights are the product of the source spectrum, refractive index of silica squared, visibility, detector efficiency and the photons transmitted by the sample. Only this last factor depends on the sample thickness.
G0 fabricated by the KNMF G1 and G2 fabricated by MicroWorks GmbH.
Nominal parameters: pitch 2.8 µm, duty cycle 0.5, absorption thickness 800 µm, phase-shifting thickness 19.4 µm for a π shift at 100 keV.
Curvature radius: G0 23 cm, G1 38.8 cm, G2 54.6 cm.
MYTHEN silicon strip detector. Custom sensor with 2 cm thickness and fan geometry of the strips.
Cospheric LLC monodisperse silica microspheres.
The images are acquired with 19 phase steps, 5 s exposure per step. Each sample is scanned 10 times with one flat scan per line, for a total of 10 flat scans. Total exposure time: 31 m 40 s per sample.
The region of interest where the microspheres are located is then manually selected after the reconstruction. Only this region of interest is shown here and considered for the analysis.